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</style></head><body><div class = "content"><div class = 'SectionBlock containment active'><div class = "S1"><span class = "S2"><span class="S3">Equivalent nodal forces</span></span></div><div class = "S1"><span class = "S2"><span class="S0">As an example, we calculate the first component of the vector of equivalent nodal forces </span></span><span class = "S2"><span class="S0"> </span></span><span class = "S2"><span class="S3">F</span></span><span class = "S2"><span class="S0"> due to a trapezoidal load with intensities </span></span><span class = "S2"><span class="S0"> </span></span><span class = "S2"><span class="S4">p1</span></span><span class = "S2"><span class="S0"> and </span></span><span class = "S2"><span class="S0"> </span></span><span class = "S2"><span class="S4">p2</span></span><span class = "S2"><span class="S0"> .</span></span></div><div class = 'CodeBlock contiguous'><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S8">%clear workspace</span></span></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0">clear </span><span class="S9">all</span><span class="S0">;</span></span></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S8">%clear command window</span></span></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0">clc;</span></span></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S8">%clear figures</span></span></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0">clf;</span></span></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0">syms </span><span class="S9">z</span></span></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0">syms </span><span class="S9">X L</span></span></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0">L=1;</span></span></div></div><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">N_1 = 1 - 3 * z^2 + 2 * z^3</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsSymbolicElement" uid="2F804EA6" data-testid="output_0" data-width="369" data-height="28" style="width: 399px; max-height: 261px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="symbolicElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span class="embeddedOutputsVariableElement" style="white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">N_1 =&nbsp;</span><span class="MathEquation inlineSymbolicElement" style="font-size: 15px; white-space: nowrap; font-style: normal; color: rgb(64, 64, 64);"><span style="vertical-align: -5px;"><img src="" width="87" height="19"></span></span></div></div></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0">figure</span></span></div></div><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">fplot(N_1, [0, 1])</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsFigure" uid="EF14235B" data-testid="output_1" style="width: 399px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="figureElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="figureContainingNode" style="width: 560px; max-width: 100%; display: inline-block; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="GraphicsView" data-dojo-attach-point="graphicsViewNode,backgroundColorNode" id="dijit__TemplatedMixin_20" widgetid="dijit__TemplatedMixin_20" style="width: 100%; height: auto; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><img class="ImageView figureImage" data-dojo-attach-point="imageViewNode" draggable="false" id="dijit__TemplatedMixin_23" widgetid="dijit__TemplatedMixin_23" src="" style="width: 100%; height: auto; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0"></span></span></div></div><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">N_3 = 3 * z^2 - 2 * z^3</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsSymbolicElement" uid="ADB6C355" data-testid="output_2" data-width="369" data-height="28" style="width: 399px; max-height: 261px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="symbolicElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span class="embeddedOutputsVariableElement" style="white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">N_3 =&nbsp;</span><span class="MathEquation inlineSymbolicElement" style="font-size: 15px; white-space: nowrap; font-style: normal; color: rgb(64, 64, 64);"><span style="vertical-align: -5px;"><img src="" width="62.5" height="19"></span></span></div></div></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0">figure</span></span></div></div><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">fplot(N_3, [0, 1])</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsFigure" uid="16A99F1E" data-testid="output_3" style="width: 399px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="figureElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="figureContainingNode" style="width: 560px; max-width: 100%; display: inline-block; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="GraphicsView" data-dojo-attach-point="graphicsViewNode,backgroundColorNode" id="dijit__TemplatedMixin_21" widgetid="dijit__TemplatedMixin_21" style="width: 100%; height: auto; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><img class="ImageView figureImage" data-dojo-attach-point="imageViewNode" draggable="false" id="dijit__TemplatedMixin_25" widgetid="dijit__TemplatedMixin_25" src="" style="width: 100%; height: auto; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div></div></div><div class = "S0"></div><div class = 'SectionBlock containment'><h2 class = "S11"><span class = "S2"><span class="S0"></span></span></h2><div class = 'CodeBlock contiguous'><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">N_2 = L * z * (z - 1)^2</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsSymbolicElement" uid="E8200C93" data-testid="output_4" data-width="369" data-height="28" style="width: 399px; max-height: 261px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="symbolicElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span class="embeddedOutputsVariableElement" style="white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">N_2 =&nbsp;</span><span class="MathEquation inlineSymbolicElement" style="font-size: 15px; white-space: nowrap; font-style: normal; color: rgb(64, 64, 64);"><span style="vertical-align: -5px;"><img src="" width="61.5" height="19"></span></span></div></div></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0">figure</span></span></div></div><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">fplot(N_2, [0, 1])</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsFigure" uid="BABCC08E" data-testid="output_5" style="width: 399px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="figureElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="figureContainingNode" style="width: 560px; max-width: 100%; display: inline-block; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="GraphicsView" data-dojo-attach-point="graphicsViewNode,backgroundColorNode" id="dijit__TemplatedMixin_26" widgetid="dijit__TemplatedMixin_26" style="width: 100%; height: auto; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><img class="ImageView figureImage" data-dojo-attach-point="imageViewNode" draggable="false" id="dijit__TemplatedMixin_29" widgetid="dijit__TemplatedMixin_29" src="" style="width: 100%; height: auto; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0">N_4=-L*(z^2-z^3);</span></span></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0">figure</span></span></div></div><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">fplot(N_4, [0, 1])</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsFigure" uid="5BC4372A" data-testid="output_6" style="width: 399px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="figureElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="figureContainingNode" style="width: 560px; max-width: 100%; display: inline-block; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="GraphicsView" data-dojo-attach-point="graphicsViewNode,backgroundColorNode" id="dijit__TemplatedMixin_27" widgetid="dijit__TemplatedMixin_27" style="width: 100%; height: auto; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><img class="ImageView figureImage" data-dojo-attach-point="imageViewNode" draggable="false" id="dijit__TemplatedMixin_31" widgetid="dijit__TemplatedMixin_31" src="" style="width: 100%; height: auto; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"></div></div></div></div></div></div></div></div><div class = "S0"></div><div class = 'SectionBlock containment'><h2 class = "S11"><span class = "S2"><span class="S0"></span></span></h2></div><div class = "S0"></div><div class = 'SectionBlock containment'><div class = "S1"><span class = "S2"><span class="S0">Works in real coordinate</span></span></div><div class = 'CodeBlock contiguous'><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0"></span></span></div></div><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">N_1 = subs(N_1, z, X/L)</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsSymbolicElement" uid="B7982A84" data-testid="output_7" data-width="369" data-height="28" style="width: 399px; max-height: 261px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="symbolicElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span class="embeddedOutputsVariableElement" style="white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">N_1 =&nbsp;</span><span class="MathEquation inlineSymbolicElement" style="font-size: 15px; white-space: nowrap; font-style: normal; color: rgb(64, 64, 64);"><span style="vertical-align: -5px;"><img src="" width="97" height="19"></span></span></div></div></div></div><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">N_2 = subs(N_2, z, X/L)</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsSymbolicElement" uid="36BF4AD0" data-testid="output_8" data-width="369" data-height="28" style="width: 399px; max-height: 261px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="symbolicElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span class="embeddedOutputsVariableElement" style="white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">N_2 =&nbsp;</span><span class="MathEquation inlineSymbolicElement" style="font-size: 15px; white-space: nowrap; font-style: normal; color: rgb(64, 64, 64);"><span style="vertical-align: -5px;"><img src="" width="71" height="19"></span></span></div></div></div></div><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">N_3 = subs(N_3, z, X/L)</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsSymbolicElement" uid="9201D0F8" data-testid="output_9" data-width="369" data-height="28" style="width: 399px; max-height: 261px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="symbolicElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span class="embeddedOutputsVariableElement" style="white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">N_3 =&nbsp;</span><span class="MathEquation inlineSymbolicElement" style="font-size: 15px; white-space: nowrap; font-style: normal; color: rgb(64, 64, 64);"><span style="vertical-align: -5px;"><img src="" width="72.5" height="19"></span></span></div></div></div></div><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">N_4 = subs(N_4, z, X/L)</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsSymbolicElement" uid="62EA1656" data-testid="output_10" data-width="369" data-height="28" style="width: 399px; max-height: 261px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="symbolicElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><span class="embeddedOutputsVariableElement" style="white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">N_4 =&nbsp;</span><span class="MathEquation inlineSymbolicElement" style="font-size: 15px; white-space: nowrap; font-style: normal; color: rgb(64, 64, 64);"><span style="vertical-align: -5px;"><img src="" width="51.5" height="19"></span></span></div></div></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0"></span></span></div></div><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">int(N_1, X)</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsSymbolicElement" uid="423A0643" data-testid="output_11" data-width="369" data-height="62" style="width: 399px; max-height: 261px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="symbolicElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="embeddedOutputsVariableElement" style="white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">ans =&nbsp;</div><span class="MathEquation displaySymbolicElement" style="font-size: 15px; white-space: nowrap; font-style: normal; color: rgb(64, 64, 64);"><span style="vertical-align: -15px;"><img src="" width="79" height="34.5"></span></span></div></div></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0"></span></span></div></div><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">int(N_1, X, sym(0), L)</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsSymbolicElement" uid="CA648794" data-testid="output_12" data-width="369" data-height="62" style="width: 399px; max-height: 261px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="symbolicElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="embeddedOutputsVariableElement" style="white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">ans =&nbsp;</div><span class="MathEquation displaySymbolicElement" style="font-size: 15px; white-space: nowrap; font-style: normal; color: rgb(64, 64, 64);"><span style="vertical-align: -15px;"><img src="" width="10.5" height="34"></span></span></div></div></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0"></span></span></div></div><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">N = [N_1; N_2; N_3; N_4]</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsSymbolicElement" uid="AD5D8DF7" data-testid="output_13" data-width="369" data-height="113" style="width: 399px; max-height: 261px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="symbolicElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="embeddedOutputsVariableElement" style="white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">N =&nbsp;</div><span class="MathEquation displaySymbolicElement" style="font-size: 15px; white-space: nowrap; font-style: normal; color: rgb(64, 64, 64);"><span style="vertical-align: -37px;"><img src="" width="123" height="84.5"></span></span></div></div></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0"></span></span></div></div><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">int(N, X)</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsSymbolicElement" uid="92EFD192" data-testid="output_14" data-width="369" data-height="184" style="width: 399px; max-height: 261px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="symbolicElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="embeddedOutputsVariableElement" style="white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">ans =&nbsp;</div><span class="MathEquation displaySymbolicElement" style="font-size: 15px; white-space: nowrap; font-style: normal; color: rgb(64, 64, 64);"><span style="vertical-align: -73px;"><img src="" width="150.5" height="155.5"></span></span></div></div></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0"></span></span></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0">syms </span><span class="S9">Feq</span></span></div></div><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">Feq</span><span class="S0"> = symhold(</span><span class="S9">'p1*int(N, X, 0, L) + (p2 - p1)*int((X*N)/L, X, 0, L)'</span><span class="S0">)</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsSymbolicElement" uid="9574F5E2" data-testid="output_15" data-width="369" data-height="65" style="width: 399px; max-height: 261px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="symbolicElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="embeddedOutputsVariableElement" style="white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">Feq =&nbsp;</div><span class="MathEquation displaySymbolicElement" style="font-size: 15px; white-space: nowrap; font-style: normal; color: rgb(64, 64, 64);"><span style="vertical-align: -15px;"><img src="" width="205" height="37.5"></span></span></div></div></div></div></div></div><div class = "S0"></div><div class = 'SectionBlock containment'><div class = "S1"><span class = "S2"><span class="S0">Finally</span></span></div><div class = 'CodeBlock contiguous'><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0">syms </span><span class="S9">p1 p2</span></span></div></div><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">Feq = p1 * int(N, X, sym(0), L) + (p2 - p1) * int((X * N)/L, X, sym(0), L)</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsSymbolicElement" uid="0B407439" data-testid="output_16" data-width="369" data-height="192" style="width: 399px; max-height: 261px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="symbolicElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="embeddedOutputsVariableElement" style="white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">Feq =&nbsp;</div><span class="MathEquation displaySymbolicElement" style="font-size: 15px; white-space: nowrap; font-style: normal; color: rgb(64, 64, 64);"><span style="vertical-align: -77px;"><img src="" width="92" height="164"></span></span></div></div></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0"></span></span></div></div><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">simplify(Feq)</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsSymbolicElement" uid="E7F71DC5" data-testid="output_17" data-width="369" data-height="192" style="width: 399px; max-height: 261px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="symbolicElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="embeddedOutputsVariableElement" style="white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">ans =&nbsp;</div><span class="MathEquation displaySymbolicElement" style="font-size: 15px; white-space: nowrap; font-style: normal; color: rgb(64, 64, 64);"><span style="vertical-align: -77px;"><img src="" width="92" height="164"></span></span></div></div></div></div></div></div><div class = "S0"></div><div class = 'SectionBlock containment'><div class = "S1"><span class = "S2"><span class="S0">Special case were p1=p2=p</span></span></div><div class = 'CodeBlock contiguous'><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0">syms </span><span class="S9">p</span></span></div></div><div class = 'inlineWrapper outputs'><div class = "S6 lineNode"><span class = "S7"><span class="S0">Feq = subs(subs(Feq, p1, p), p2, p)</span></span></div><div class="outputParagraph" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 13.9999990463257px;"><div class="inlineElement eoOutputWrapper embeddedOutputsSymbolicElement" uid="4A19EC2C" data-testid="output_18" data-width="369" data-height="180" style="width: 399px; max-height: 261px; white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="symbolicElement" style="white-space: normal; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;"><div class="embeddedOutputsVariableElement" style="white-space: pre-wrap; font-style: normal; color: rgb(64, 64, 64); font-size: 12px;">Feq =&nbsp;</div><span class="MathEquation displaySymbolicElement" style="font-size: 15px; white-space: nowrap; font-style: normal; color: rgb(64, 64, 64);"><span style="vertical-align: -71px;"><img src="" width="57" height="152"></span></span></div></div></div></div></div></div><div class = "S0"></div><div class = 'SectionBlock containment'><div class = "S1"><span class = "S2"><span class="S0"> That's all</span></span></div></div><div class = "S0"></div><div class = 'SectionBlock containment'><h2 class = "S11"><span class = "S2"><span class="S0">Local Functions</span></span></h2><h2 class = "S12"><span class = "S2"><span class="S0">Helpers</span></span></h2><div class = 'CodeBlock contiguous'><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S13">function </span><span class="S0">h = symhold(exprstring)</span></span></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S0">h = evalin(symengine, [</span><span class="S9">'hold(' </span><span class="S0">exprstring </span><span class="S9">' )'</span><span class="S0">]);</span></span></div></div><div class = 'inlineWrapper'><div class = "S6 lineNode"><span class = "S7"><span class="S13">end</span></span></div></div></div></div></div>
<!-- 
##### SOURCE BEGIN #####
%% 
% *Equivalent nodal forces*
% 
% As an example, we calculate the first component of the vector of equivalent 
% nodal forces  *F* due to a trapezoidal load with intensities  _p1_ and  _p2_ 
% .

%clear workspace
clear all;
%clear command window
clc;
%clear figures
clf;
syms z
syms X L
L=1;
N_1 = 1 - 3 * z^2 + 2 * z^3
figure
fplot(N_1, [0, 1])

N_3 = 3 * z^2 - 2 * z^3
figure
fplot(N_3, [0, 1])
%% 
%%
N_2 = L * z * (z - 1)^2
figure
fplot(N_2, [0, 1])
N_4=-L*(z^2-z^3);
figure
fplot(N_4, [0, 1])
%% 
% Works in real coordinate
%%

N_1 = subs(N_1, z, X/L)
N_2 = subs(N_2, z, X/L)
N_3 = subs(N_3, z, X/L)
N_4 = subs(N_4, z, X/L)

int(N_1, X)

int(N_1, X, sym(0), L)

N = [N_1; N_2; N_3; N_4]

int(N, X)

syms Feq
Feq = symhold('p1*int(N, X, 0, L) + (p2 - p1)*int((X*N)/L, X, 0, L)')
%% 
% Finally
%%
syms p1 p2
Feq = p1 * int(N, X, sym(0), L) + (p2 - p1) * int((X * N)/L, X, sym(0), L)

simplify(Feq)
%% 
% Special case were p1=p2=p
%%
syms p
Feq = subs(subs(Feq, p1, p), p2, p)
%% 
%  That's all
%% Local Functions
%% Helpers
%%
function h = symhold(exprstring)
h = evalin(symengine, ['hold(' exprstring ' )']);
end
##### SOURCE END #####
--></body></html>